Interpolating polynomial given only Y values

Question

Can we reconstruct a polynomial with only Y values? What if the number of Y values are far more than the degree of the polynomial? Also can we obtain the root of this polynomial with this Y's value without interpolating?(i.e. without knowing the coefficients of the polynomial)

Answer 1

If we only have a set of values (which, as you defined, we can call Y-values) with no relation to another set values, there is no singular meaningful and unique way to create a function to interpolate them. This does not just apply to polynomials.

So no, if we are given simply a set of Y-values and told that they appear as certain $f(x)$ values in a polynomial, we do not have enough information to narrow the possibilities down to a single polynomial (and not even to a single power). The number of options we have is infinite. All we get to know, when we are given such a list, is that the range of the polynomial must at least have a global minimum or maximum corresponding the the smallest and largest y-values in the set of the presented points, if we are also told that its power is even. The polynomial may also transgress both the upper and lower y-values if its power is odd.

If you are unconvinced, consider the following: ones gives us an arbitrary (even infinite) list of y-values and says that they appear as $f(x)$s on a polynomial. Now we are asked what the polynomial is.

We can always answer $x=y$ and we will be right, not matter what the list (as long as we are just given y-values).

Regarding your final point, if we can't derive any meaningful information about the nature of the polynomial (as it really could be just about anything with the exception of certain even powered polynomials), it is not very meaningful to solve for a root, as the root could be anything.

For further reading, I would recommend you look up "functions" and what is means to form a "function".

Does hesitate to ask for clarification if something doesn't make sense.

Have a nice day :)

Answer 2

I interpret the question as: given a set $\{y_1,\dots,y_n\}$ of values of an unknown polynomial, can one reconstruct the polynomial? The answer is an emphatic no: every polynomial of odd degree takes all real values, for example (and many polynomials of even degree will also be consistent with whatever finite data you have).

If you know that the values in question are taken in that order (so, there exist $x_1<x_2<\cdots x_n$ such that $f(x_j)=y_j$), then you can at least say that the degree cannot be too small (it has to be greater than the number of interior local extrema of the sequence $(y_1,\dots,y_n)$. But there's not much else you're going to be able to say.